Integral Representations of Certain Measures in the One-Dimensional Diffusions Excursion Theory

In this note we present integral representations of the Itô excursion measure associated with a general one-dimensional diffusion X. These representations and identities are natural extensions of the classical ones for reflected Brownian motion, RBM. As is well known, the three-dimensional Bessel process, BES(3), plays a crucial rôle in the analysis of the Brownian excursions. Our main interest is in showing explicitly how certain excursion theoretical formulae associated with the pair (RBM, BES(3)) generalize to pair (X, X ^↑ ), where X ^↑ denotes the diffusion obtained from X by conditioning X not to hit 0. We illustrate the results for the pair \((R_{-},R_{+})\) consisting of a recurrent Bessel process with dimension \(d_{-} = 2(1-\alpha ),\) α ∈ (0, 1), and a transient Bessel process with dimension \(d_{+} = 2(1+\alpha )\). Pair (RBM, BES(3)) is, clearly, obtained by choosing \(\alpha = 1/2.\)